It's Relative II

 

Dead Man's Curve

(Dead Man's Curve) is no place to play
(Dead Man's Curve) you'd best keep away
(Dead Man's Curve) I can hear 'em say
Won't come back from Dead Man's Curve

Has relativity thrown us for a curve?

As anyone who has traveled long distances by air knows, the shortest distance between two points on a sphere is NOT a straight line as would be suggested by Pythagoras’ Theorem. 

c²=a²+b²

c=sqrt(a²+b² )

This theorem would suggest that for a right triangle, with sides a and side b, the shortest line between these two points would be the hypotenuse, c.  However, for large distances on the earth, this would require leaving the surface of the Earth and crashing into and travelling through the Earth.  If travel is confined to the surface, or just above the surface, traveling over the sphere suggests that the shortest distance is a Great Circle Distance.

This is an example of non-Euclidean Geometry, where Euclidean space deals with a flat surface.  The surface of the Earth is a sphere, not flat.  The Spherical Pythagorean Theorem is

cos (c/R)=cos(a/R)*cos (b/R)

As the radius of the sphere, R,  goes to infinity, the surface of that sphere becomes virtually flat, and expanding the cosines using their Taylor series, yields  

c=sqrt(a²+b²)

which is the traditional Pythagorean Theorem.  If a, b and c are very small compared to the radius, R, of the sphere, e.g. a small surface in your home, the surface is virtually flat and Pythagoras' Theorem is virtually true.

But a surface can not only be spherical, what mathematicians would call concave.  A surface can also be a hyperbolic plane, what mathematicians would call convex.  In this case the Hyperbolic Pythagorean Theorem is

cosh(c)=cosh(a)*cosh(b)

A hyperbolic cosine can also be expressed using Euler’s Number, the exponential e, as  

cosh (x)=½ *(e^x+e^-x)

This means that the Hyperbolic Pythagorean Theorem, in exponential form, is

½*(e^c+e^-c )= ½*(e^a+e^-a)* ½*(e^b+e^-b )

which becomes

c=ln(cosh(a)cosh(b)±sinh(a)sinh(b))

In most real-world applications, the traditional Pythagorean Theorem is good enough.  When the sphere is very large, the surface of the sphere is approximately flat.  When the distance between the points is large compared to the radius of the sphere, e.g. the surface of the Earth, the Spherical Pythagorean Theorem will give the Great Circle Distance on the spherical surface.

In most applications, the Hyperbolic Pythagorean Theorem will not be needed.  However, there is a notable instance where the use of the traditional, flat, Pythagorean Theorem might lead to the wrong conclusion.

Einstein showed that the velocity of an object, v, is limited by the speed of light, c.  His famous equation, E=mc², is an implication of that limit.  The Total Energy of an object is the combination of its Kinetic Energy, related to its velocity, and its Rest Energy, related to the speed of light.  Kinetic Energy is what powers cannonballs.  The equation for Kinetic Energy is ½ mv².   Rest Energy is what powers the Atomic Bomb.  The equation for Rest Energy is m₀c², where m₀ is the rest mass at a velocity of zero.  In most real-world examples, the velocity is very small compared to the speed of light and the kinetic mass, m, is approximately equal to the rest mass, m₀.  But strictly speaking the kinetic mass is a factor, γ, multiplied by the rest mass, where that factor depends on the ratio of the velocity of the mass to the speed of light. The magnitude of the combination  makes Einstein’s triangle of energy 

(γm₀ c² )²=(½ γm₀ v² )²+(m₀ c² )²

If you solve for γ using Pythagoras’ Theorem in flat space, this leads to Lorentz's factor as

γ=sqrt(1-(v/c)² )

This has the notable problem that it leads to an undefined term, 1/0, when the velocity of an object, v,  is equal to the speed of light, c. But the use of the Traditional Pythagorean Theorem also assumes that the surface is flat.  If the surface is assumed to be hyperbolic, (space-time is known to be curved, not flat.  The surface could also be Spherical in which case the Radius, R,  of the surface, i.e. universe, would need to be used. However the Radius of space is so large that it approaches infinity in which case, the surface would be equivalent to a flat surface), this relativistic factor could be:

γ=1+ln(1-(v/c)² )

This factor has a correlation with the traditional Lorentz factor, when v is less than 95% of the speed of light, of 0.986.  It also has the characteristic of being infinite, not undefined, when the velocity is equal to the speed of light. The proposed factor differs by less than 5% from the traditional Lorentz factor for speeds less than 90% of the speed of light, and differs from the traditional Lorentz factor by less than 1% for speeds less than 23% of the speed of light.